Session: VIB-08-02: Nonlinear Systems & Phenomena II

Paper Number: 97866

97866 - Modeling and Analysis of a Constrained Two-Rigid-Body System Using Nonholonomic Lagrange Equations 

In this work we model a linked system of two rigid bodies, with a nonholonomic constraint, as a “slot car and trailer,” with a lead body called the “head” or “car” pinned in a slot, and the trailing body called the “tail” or “trailer” with a constraint representing a wheel or skate.  This system is related to models of a truck and trailer [1], Chaplygin sleigh and landfish [2].  The lead body is constrained holonomically and the trailer is constrained nonholonomically with a massless wheel or skate, such that its velocity is tangent to its orientation angle.  In the initial simplified modeling, either the head or tail angle will be imposed, and the other angle will take the role of the constrained dynamic variable. Among the three variables and two constraints, the system has one degree of freedom, with the head displacement as the independent dynamic variable.  We apply nonholonomic Lagrange (NHL) equations [3], and analyze behavior of the resulting equations of motion.

The nonholonomic constraint and the NHL equation lead to a nonautonomous second-order system. Simulations were conducted on the cases of imposed head angle and imposed tail angle.  Examples of practical regular motions are presented for the two-body system starting at rest in a horizontal configuration.  In such case, the imposed head angle leads to oscillations in the tail, and the net system velocity which builds up.  An imposed tail angle incurs oscillations in the head as again the net system velocity builds up.  A variety of simulations (to be shown in the presentation) show that for certain parameter values, regular forward motion is achievable while other parameter values may result in chaotic motion.  More investigations are needed to quantify the existence of chaos.  Ongoing studies include systematic simulations for uncovering classes of behavior, and small-parameter perturbation analyses for revealing roles of parameters in regular motion.  Follow-up studies will also involve variations in the form of the constraints, and the addition of damping.

Acknowledgement

This material is based on work supported by the National Science Foundation under grant number CMMI–2015194.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

[1] Zoebel, D., 2003, “Trajectory segmentation for the autonomous control of backward motion for truck and trailer,” IEEE Transactions on Intelligent Transportation Systems, 4 (2) 59-66.

[2] Dear, T., Kelly, S. D., Travers, M., and Choset H., 2013, “Mechanics and control of a terrestrial vehicle exploiting a nonholonomic constraint for fishlike locomotion,” Dynamic Systems and Control Conference, October 21-23, Palo Alto, DSCC2013-3941.

[3] Feeny, B. F., 2006, “D’Alembert’s principle and the equations of nonholonomic motion,” ASME IMECE’06, Nov.~5-10, Chicago, IMECE2006/VIB-14533.

 

Presenting Author: Jamal Ardister Michigan State University

Presenting Author Biography: Jamal Ardister is a Ph.D. student at Michigan State University studying mechanical engineering under the advisement of Dr. Brian Feeny. Jamal specializes in nonlinear vibrations and nonholonomic systems.

Authors:

Jamal Ardister Michigan State University
Brian Feeny Michigan State University